Lebesgue property of convex risk measures for bounded càdlàg processes (Q390190)

For a financial position, the supervising authority determines the proprietary capital needed to secure the liabilities. The concept of coherent risk measures was introduced by \textit{P. Artzner} et al. [Math. Finance 9, No. 3, 203--228 (1999; Zbl 0980.91042)]. \textit{H. Föllmer} and \textit{A. Schied} [in: K. Sandmann (ed.) et al., Advances in finance and stochastics. Essays in honour of Dieter Sondermann. Berlin: Springer. 39--56 (2002; Zbl 1022.91045)] generalised the concept to convex risk measures. An important property of risk measures is the Fatou property. If a sequence of risks \(\{X_n\}\) converges to a risk \(X\) in probability, then it should hold for a risk measure \(\rho\) that \(\rho(X) \leq \lim_{n \to \infty} \rho(X_n)\). \textit{P. Cheridito} et al. [Stochastic Processes Appl. 112, No. 1, 1--22 (2004; Zbl 1114.91047)] gave a dual representation for the Fatou property. This duality concept is generalised. A risk measure has the Lebesgue property if for any bounded sequence \(\{X_n\}\) that converges to a risk \(X\) it holds that \(\lim_{n \to \infty} \rho(X_n) = \rho(X)\). The main result are ten equivalent characterisations for the Lebesgue property.